New improvements of Jensen’s type inequalities via 4-convex functions with applications

In this article, we present some new improvements of Jensen’s type inequalities via 4-convex and Green functions. These improvements are demonstrated in discrete as well as in integral versions. The aforesaid results enable us to give some improvements of Jensen’s and the Jensen–Steffensen inequalities. Also, we present some improvements of the reverse Jensen’s and the Jensen–Steffensen inequalities. Then as consequences of the improved Jensen’s inequality, we deduce some new bounds for the power, geometric and quasi-arithmetic means, also obtain bounds for the Hermite–Hadamard gap and improvements of the Hölder inequality. Finally as applications of the improved Jensen’s inequality, we present some new bounds for various divergences and Zipf–Mandelbrot entropy. © 2021, The Royal Academy of Sciences, Madrid.